# Properties

 Label 768.4.c.l Level $768$ Weight $4$ Character orbit 768.c Analytic conductor $45.313$ Analytic rank $0$ Dimension $4$ CM discriminant -8 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,4,Mod(767,768)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("768.767");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 768.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$45.3134668844$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 24) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 3 \beta_{2} + 2 \beta_1) q^{3} + (5 \beta_{3} - 23) q^{9}+O(q^{10})$$ q + (-3*b2 + 2*b1) * q^3 + (5*b3 - 23) * q^9 $$q + ( - 3 \beta_{2} + 2 \beta_1) q^{3} + (5 \beta_{3} - 23) q^{9} + ( - 25 \beta_{2} - 25 \beta_1) q^{11} - 38 \beta_{3} q^{17} + (53 \beta_{2} - 53 \beta_1) q^{19} + 125 q^{25} + (84 \beta_{2} - 11 \beta_1) q^{27} + (125 \beta_{3} + 100) q^{33} + 20 \beta_{3} q^{41} + (145 \beta_{2} - 145 \beta_1) q^{43} + 343 q^{49} + ( - 114 \beta_{2} - 266 \beta_1) q^{51} + ( - 53 \beta_{3} + 530) q^{57} + ( - 115 \beta_{2} - 115 \beta_1) q^{59} + (35 \beta_{2} - 35 \beta_1) q^{67} + 430 q^{73} + ( - 375 \beta_{2} + 250 \beta_1) q^{75} + ( - 230 \beta_{3} + 329) q^{81} + (241 \beta_{2} + 241 \beta_1) q^{83} + 470 \beta_{3} q^{89} + 1910 q^{97} + (75 \beta_{2} + 1075 \beta_1) q^{99}+O(q^{100})$$ q + (-3*b2 + 2*b1) * q^3 + (5*b3 - 23) * q^9 + (-25*b2 - 25*b1) * q^11 - 38*b3 * q^17 + (53*b2 - 53*b1) * q^19 + 125 * q^25 + (84*b2 - 11*b1) * q^27 + (125*b3 + 100) * q^33 + 20*b3 * q^41 + (145*b2 - 145*b1) * q^43 + 343 * q^49 + (-114*b2 - 266*b1) * q^51 + (-53*b3 + 530) * q^57 + (-115*b2 - 115*b1) * q^59 + (35*b2 - 35*b1) * q^67 + 430 * q^73 + (-375*b2 + 250*b1) * q^75 + (-230*b3 + 329) * q^81 + (241*b2 + 241*b1) * q^83 + 470*b3 * q^89 + 1910 * q^97 + (75*b2 + 1075*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 92 q^{9}+O(q^{10})$$ 4 * q - 92 * q^9 $$4 q - 92 q^{9} + 500 q^{25} + 400 q^{33} + 1372 q^{49} + 2120 q^{57} + 1720 q^{73} + 1316 q^{81} + 7640 q^{97}+O(q^{100})$$ 4 * q - 92 * q^9 + 500 * q^25 + 400 * q^33 + 1372 * q^49 + 2120 * q^57 + 1720 * q^73 + 1316 * q^81 + 7640 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$-\zeta_{8}^{3} - \zeta_{8}^{2} + \zeta_{8}$$ -v^3 - v^2 + v $$\beta_{2}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}$$ -v^3 + v^2 + v $$\beta_{3}$$ $$=$$ $$2\zeta_{8}^{3} + 2\zeta_{8}$$ 2*v^3 + 2*v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} + \beta_1 ) / 4$$ (b3 + b2 + b1) / 4 $$\zeta_{8}^{2}$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 2$$ (b2 - b1) / 2 $$\zeta_{8}^{3}$$ $$=$$ $$( \beta_{3} - \beta_{2} - \beta_1 ) / 4$$ (b3 - b2 - b1) / 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$$.

 $$n$$ $$257$$ $$511$$ $$517$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
767.1
 0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i
0 −1.41421 5.00000i 0 0 0 0 0 −23.0000 + 14.1421i 0
767.2 0 −1.41421 + 5.00000i 0 0 0 0 0 −23.0000 14.1421i 0
767.3 0 1.41421 5.00000i 0 0 0 0 0 −23.0000 14.1421i 0
767.4 0 1.41421 + 5.00000i 0 0 0 0 0 −23.0000 + 14.1421i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.4.c.l 4
3.b odd 2 1 inner 768.4.c.l 4
4.b odd 2 1 inner 768.4.c.l 4
8.b even 2 1 inner 768.4.c.l 4
8.d odd 2 1 CM 768.4.c.l 4
12.b even 2 1 inner 768.4.c.l 4
16.e even 4 1 24.4.f.a 2
16.e even 4 1 96.4.f.a 2
16.f odd 4 1 24.4.f.a 2
16.f odd 4 1 96.4.f.a 2
24.f even 2 1 inner 768.4.c.l 4
24.h odd 2 1 inner 768.4.c.l 4
48.i odd 4 1 24.4.f.a 2
48.i odd 4 1 96.4.f.a 2
48.k even 4 1 24.4.f.a 2
48.k even 4 1 96.4.f.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.4.f.a 2 16.e even 4 1
24.4.f.a 2 16.f odd 4 1
24.4.f.a 2 48.i odd 4 1
24.4.f.a 2 48.k even 4 1
96.4.f.a 2 16.e even 4 1
96.4.f.a 2 16.f odd 4 1
96.4.f.a 2 48.i odd 4 1
96.4.f.a 2 48.k even 4 1
768.4.c.l 4 1.a even 1 1 trivial
768.4.c.l 4 3.b odd 2 1 inner
768.4.c.l 4 4.b odd 2 1 inner
768.4.c.l 4 8.b even 2 1 inner
768.4.c.l 4 8.d odd 2 1 CM
768.4.c.l 4 12.b even 2 1 inner
768.4.c.l 4 24.f even 2 1 inner
768.4.c.l 4 24.h odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(768, [\chi])$$:

 $$T_{5}$$ T5 $$T_{7}$$ T7 $$T_{11}^{2} - 5000$$ T11^2 - 5000 $$T_{13}$$ T13 $$T_{23}$$ T23

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 46T^{2} + 729$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$(T^{2} - 5000)^{2}$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + 11552)^{2}$$
$19$ $$(T^{2} + 11236)^{2}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$(T^{2} + 3200)^{2}$$
$43$ $$(T^{2} + 84100)^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$(T^{2} - 105800)^{2}$$
$61$ $$T^{4}$$
$67$ $$(T^{2} + 4900)^{2}$$
$71$ $$T^{4}$$
$73$ $$(T - 430)^{4}$$
$79$ $$T^{4}$$
$83$ $$(T^{2} - 464648)^{2}$$
$89$ $$(T^{2} + 1767200)^{2}$$
$97$ $$(T - 1910)^{4}$$